ALGEBRA. A branch of pure mathematic that materially simplifies the solution of arith metical problems, especially through the use of equations. It also forms the introduction to all of the higher branches of mathematical science, except pure geometry.
The lame is derived from the title of the Arabic work by Al-Khuwarizmi (q.v.), Dm al jabr mugabalah, meaning "the science of re diutegration and equation;" that is, the science that relates to the reduction of equations to in tegral form and to the transposition of terms. The title appeared thereafter in various forms, as ludas algebra' ohnugrabakegue, and algiebar and almachabel, but the abbreviation algebra was finally adopted. The science also went under various other names in the fifteenth and sixteenth centuries, as the ars magna (Carlin, 1545), the uric ntoggiorc, the rego/a de la cosy (because the unknown quantity was denominated cosa, the "thing"), and hence in early English the cossike art, and in German the Coss.
The exact limitations of algebra are not gener ally agreed upon by mathematicians, and hence various definitions have been proposed for the science. It has been proposed to limit it to the theory of equations, as the etymology of the word would suggest; but this has become a separate branch of mathematics. Perhaps the most satis factory definition, especially as it brings out the distinction between algebra and arithmetic, is that of Comte: "Algebra is the calculus of func tions. and arithmetic is the calculus of values." This distinction would include some arithmetic in ordinary school algebra (e. g., the study of surds), and some algebra in common arithmetic (e.g.. the formula for square root).
The oldest known manuscript in which al gebra is treated is that of Ahmes, the Egyptian scribe, who, about 1700 B.C., copied a treatise dating perhaps from 2500 B.C. In this ap pears the simple equation in the form, "Han (literally heap), its seventh, its whole, it makes 19." which, put in modern symbols, means = 19. In Euclid's Men/cats (about 300 B.c.) a knowledge of certain quadratic equations is shown. It was Diophantus of Alexandria (q.v.), however, who made the first attempt (fourth century A.D.) to work out the science. In the fol lowing century Aryabhatta (q.v.) made some contributions to the subject. Little was then done until about S00 A.D., when Al-Khuwarizmi wrote. His efforts were followed by another period of comparative repose, until the Italian algehraists of the sixteenth century undertook the solution of the cubic equation. (See EquA -"lox.) In this, building upon the efforts of Ferreo and Tartaglia, Cardan was successful (1545), although there is reason to believe that the real honor belongs to Tartaglia. Soon after, Ferrari and Bombelli (1579) gave the solution of the biquadratie equation.
The principal improvements in the succeeding century related to symbolism. It took a long time, however, to pass from the radical sign of Chuquet (1484), 10 through various forms, as 4/T110, to our common symbol VII) and to the more refined Similarly it was only by slow steps that progress was made from Cardan's eebus p 6 rebus iequalis 20, for 6x = 20, through Vieta's